1.4 The sum of the \( n \) terms of a sequence is given by \[ S_{n}=\frac{n}{2}(7 n+15) \] Calculate \( T_{4} \)

To find the 4th term (\( T_4 \)) of the sequence, we can use the formula for the sum of the first \( n \) terms (\( S_n \)): \[ S_n = \frac{n}{2}(7n + 15) \] The \( n \)-th term of the sequence (\( T_n \)) can be found by subtracting the sum of the first \( n-1 \) terms from the sum of the first \( n \) terms: \[ T_n = S_n - S_{n-1} \] Let's calculate \( T_4 \): 1. **Calculate \( S_4 \):** \[ S_4 = \frac{4}{2}(7 \times 4 + 15) = 2(28 + 15) = 2 \times 43 = 86 \] 2. **Calculate \( S_3 \):** \[ S_3 = \frac{3}{2}(7 \times 3 + 15) = \frac{3}{2}(21 + 15) = \frac{3}{2} \times 36 = 54 \] 3. **Find \( T_4 \):** \[ T_4 = S_4 - S_3 = 86 - 54 = 32 \] **Alternatively**, you can derive a general formula for \( T_n \) directly: \[ T_n = 7n + 4 \] Plugging in \( n = 4 \): \[ T_4 = 7 \times 4 + 4 = 28 + 4 = 32 \] **Answer:** \( T_{4} = 32 \)